Consider the case of the following annuities, and the need to compute either the

Question

Consider the case of the following annuities, and the need to compute either their expected rate of return or duration. Joshua needed money for some unexpected expenses, so he borrowed $5,355.26 from a friend and agreed to repay the loan in seven equal installments of $1,100 at the end of each year. The agreement is offering an implied interest rate of Joshua's friend, Willie, has hired a financial planner for advice on retirement. Considering Willie's current expenses and expected future lifestyle changes, the financial planner has stated that once Willie crosses a threshold of $12,171,957 in savings, he will have enough money for retirement. Willie has nothing saved for his retirement yet, so he plans to start depositing $70,000 in a retirement fund at a fixed rate of 10.00% at the end of each year. It will take years for Willie to reach his retirement goal

Explanation / Answer

Question - 1

1100 * PVIFA = 5355.26

PVIFA = 5355.26 / 1100 = 4.868418

PVIFA = [ 1 - (1+r)-n ] / r = 4.868418

[ 1 - ( 1 + r )-7 ] / r = 4.868418

Using trail and error method, we find that above equation is satisfied for ....... r = 10%

Hence ........ implied interest rate = 10%

OR

In an excel file............ A1, A2,A3,A4,A5,A6,A7 and A8 ........ be entered with values = -5355.26, 1100,1100,1100,1100,1100,1100 and 1100.

Now ......... enter in any cell ......... the formula ............ " = IRR(A1:A8) " ........ Press enter to get the value = 10 %

Question - 2

Here 70000 is the annual deposit ...... so it is annuity

n = number of years .......... to be calculated

r = 0.10

Annuity * FVIFA = Future value

70000 * [ (1.10)n - 1 ] / 0.10 = 12,171,957

[ (1.10)n - 1 ] = 12,171,957 * 0.10 / 70000 = 17.38851

(1.10)n = 17.38851 + 1 = 18.38851

n * Log (1.10) = Log (18.38851)

n * 0.0413926852 = 1.26454654

n = 1.26454654 / 0.0413926852 = 30.55 years

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